Question

show that root 2 is an irrational

Answer 1

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Answer 2

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Let us assume , that the √2 is rational....

So, We can find integer p and q.. (where q ≠ 0)

Such that ,

$\sqrt{2} = \frac{p}{q}$

Suppose p and q have a common factor other than 1 . Then we divide by the Common factor to get ...

$\sqrt{2} = \frac{a}{b}$
Where a and b are Comprime .

So, b√2 = a ..


Squaring both side ..

(b√2)² = a²

2b² = a²

Therefore , 2 divides a²

So if 2 divides a² then 2 divides a also ...

So we can write ..

a = 2c ...for some integer c....


Substituting for a ,

we get ..

2b² = 4c² . That is b² = 2c²

..

This means 2 divides a² and so 2. divides b ...

Therefore a nad b..have at least 2 common factor..

But this Contradict ..that a and b have no common factor other than 1 ..


So , our assuming is wrong ...

we conclude that √2 is irrational Number ...


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